How to find the closed form definition of a series?

Question

How does one determine if the following series is arithmetic or geometric. If it is arithmetic, what is the common difference? If it is geometric, what is the common ratio? Using this information, find the closed-form definition of the series.

 

n  f(n)  Σ
0  1     1
1  5     6
2  14   20 
3  30   50
4  55   105

Answer 1

In an arithmetic sequence the terms have a common difference: a_n+1 - a_n = d

In a geometric sequence the terms have a common ratio: a_n+1 / a_n = r

 

5 - 1 = 4

14 - 5 = 9

30 - 14 = 16

55 - 30 = 25

 

The series is neither arithmetic nor geometric.

 

a_n+1 = a_n + (n + 1)²